4 research outputs found
Heredity for generalized power domination
In this paper, we study the behaviour of the generalized power domination
number of a graph by small changes on the graph, namely edge and vertex
deletion and edge contraction. We prove optimal bounds for
, and for in
terms of , and give examples for which these bounds are
tight. We characterize all graphs for which for any edge . We also consider the behaviour of the
propagation radius of graphs by similar modifications.Comment: Discrete Mathematics and Theoretical Computer Science, 201
Power Domination Number On Shackle Operation with Points as Lingkage
The Power dominating set is a minimum point of determination in a graph that can dominate the connected dots around it, with a minimum domination point. The smallest cardinality of a power dominating set is called a power domination number with the notation . The purpose of this study is to determine the Shackle operations graph value from several special graphs with a point as a link. The result operation graphs are: Shackle operation graph from Path graph , Shackle operation graph from Sikel graph , Shackle operation graph from Star graph . The method used in this paper is axiomatic deductive method in solving problems. Understanding the axiomatic method itself is a method of deductive proof principles that applies in mathematical logic by using theorems that already exist in solving a problem. In this paper begins by determining the paper object that is the Shackle point operations result graph. Next, determine the cardinality of these graphs. After that, determine the point that has the maximum degree on the graph as the dominator point of power domination. Then, check whether the nearest neighbor has two or more degrees and analyze its optimization by using a ceiling function comparison between zero forching with the greatest degree of graph. Thus it can be determined ϒp minimal and dominated. The results of the power domination number study on Shackle operation graph result with points as connectors are , for  and ; , for  and ; , for  and
Restricted power domination and zero forcing problems
Power domination in graphs arises from the problem of monitoring an electric power system by placing as few measurement devices in the system as possible. A power dominating set of a graph is a set of vertices that observes every vertex in the graph, following a set of rules for power system monitoring. A practical problem of interest is to determine the minimum number of additional measurement devices needed to monitor a power network when the network is expanded and the existing devices remain in place. In this paper, we study the problem of finding the smallest power dominating set that contains a given set of vertices X. We also study the related problem of finding the smallest zero forcing set that contains a given set of vertices X. The sizes of such sets in a graph G are respectively called the restricted power domination number and restricted zero forcing number of G subject to X. We derive several tight bounds on the restricted power domination and zero forcing numbers of graphs, and relate them to other graph parameters. We also present exact and algorithmic results for computing the restricted power domination number, including integer programs for general graphs and a linear time algorithm for graphs with bounded treewidth. We also use restricted power domination to obtain a parallel algorithm for finding minimum power dominating sets in trees